Properties

Label 975.1.bi.a
Level $975$
Weight $1$
Character orbit 975.bi
Analytic conductor $0.487$
Analytic rank $0$
Dimension $16$
Projective image $D_{20}$
CM discriminant -39
Inner twists $8$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 975.bi (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.486588387317\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{40})\)
Defining polynomial: \(x^{16} - x^{12} + x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{20}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{20} - \cdots)\)

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + ( \zeta_{40}^{9} - \zeta_{40}^{15} ) q^{2} -\zeta_{40}^{14} q^{3} + ( \zeta_{40}^{4} - \zeta_{40}^{10} + \zeta_{40}^{18} ) q^{4} -\zeta_{40} q^{5} + ( \zeta_{40}^{3} - \zeta_{40}^{9} ) q^{6} + ( -\zeta_{40}^{5} - \zeta_{40}^{7} + \zeta_{40}^{13} - \zeta_{40}^{19} ) q^{8} -\zeta_{40}^{8} q^{9} +O(q^{10})\) \( q + ( \zeta_{40}^{9} - \zeta_{40}^{15} ) q^{2} -\zeta_{40}^{14} q^{3} + ( \zeta_{40}^{4} - \zeta_{40}^{10} + \zeta_{40}^{18} ) q^{4} -\zeta_{40} q^{5} + ( \zeta_{40}^{3} - \zeta_{40}^{9} ) q^{6} + ( -\zeta_{40}^{5} - \zeta_{40}^{7} + \zeta_{40}^{13} - \zeta_{40}^{19} ) q^{8} -\zeta_{40}^{8} q^{9} + ( -\zeta_{40}^{10} + \zeta_{40}^{16} ) q^{10} + ( \zeta_{40} - \zeta_{40}^{3} ) q^{11} + ( -\zeta_{40}^{4} + \zeta_{40}^{12} - \zeta_{40}^{18} ) q^{12} -\zeta_{40}^{18} q^{13} + \zeta_{40}^{15} q^{15} + ( -1 - \zeta_{40}^{2} + \zeta_{40}^{8} - \zeta_{40}^{14} - \zeta_{40}^{16} ) q^{16} + ( -\zeta_{40}^{3} - \zeta_{40}^{17} ) q^{18} + ( -\zeta_{40}^{5} + \zeta_{40}^{11} - \zeta_{40}^{19} ) q^{20} + ( \zeta_{40}^{10} - \zeta_{40}^{12} - \zeta_{40}^{16} + \zeta_{40}^{18} ) q^{22} + ( -\zeta_{40} + \zeta_{40}^{7} - \zeta_{40}^{13} + \zeta_{40}^{19} ) q^{24} + \zeta_{40}^{2} q^{25} + ( \zeta_{40}^{7} - \zeta_{40}^{13} ) q^{26} -\zeta_{40}^{2} q^{27} + ( -\zeta_{40}^{4} + \zeta_{40}^{10} ) q^{30} + ( \zeta_{40}^{3} + \zeta_{40}^{5} - \zeta_{40}^{9} - \zeta_{40}^{11} + \zeta_{40}^{15} + \zeta_{40}^{17} ) q^{32} + ( -\zeta_{40}^{15} + \zeta_{40}^{17} ) q^{33} + ( \zeta_{40}^{6} - \zeta_{40}^{12} + \zeta_{40}^{18} ) q^{36} -\zeta_{40}^{12} q^{39} + ( -1 + \zeta_{40}^{6} + \zeta_{40}^{8} - \zeta_{40}^{14} ) q^{40} + ( \zeta_{40}^{5} + \zeta_{40}^{11} ) q^{41} + ( \zeta_{40}^{8} + \zeta_{40}^{12} ) q^{43} + ( \zeta_{40} + \zeta_{40}^{5} - \zeta_{40}^{7} - \zeta_{40}^{11} + \zeta_{40}^{13} + \zeta_{40}^{19} ) q^{44} + \zeta_{40}^{9} q^{45} + ( -\zeta_{40}^{11} - \zeta_{40}^{17} ) q^{47} + ( \zeta_{40}^{2} - \zeta_{40}^{8} - \zeta_{40}^{10} + \zeta_{40}^{14} + \zeta_{40}^{16} ) q^{48} - q^{49} + ( \zeta_{40}^{11} - \zeta_{40}^{17} ) q^{50} + ( \zeta_{40}^{2} - \zeta_{40}^{8} + \zeta_{40}^{16} ) q^{52} + ( -\zeta_{40}^{11} + \zeta_{40}^{17} ) q^{54} + ( -\zeta_{40}^{2} + \zeta_{40}^{4} ) q^{55} + ( \zeta_{40}^{7} + \zeta_{40}^{9} ) q^{59} + ( \zeta_{40}^{5} - \zeta_{40}^{13} + \zeta_{40}^{19} ) q^{60} + ( -\zeta_{40}^{6} - \zeta_{40}^{18} ) q^{61} + ( 1 - \zeta_{40}^{4} - \zeta_{40}^{6} + \zeta_{40}^{10} + \zeta_{40}^{12} + \zeta_{40}^{14} - \zeta_{40}^{18} ) q^{64} + \zeta_{40}^{19} q^{65} + ( \zeta_{40}^{4} - \zeta_{40}^{6} - \zeta_{40}^{10} + \zeta_{40}^{12} ) q^{66} + ( \zeta_{40}^{13} - \zeta_{40}^{15} ) q^{71} + ( \zeta_{40} - \zeta_{40}^{7} + \zeta_{40}^{13} + \zeta_{40}^{15} ) q^{72} -\zeta_{40}^{16} q^{75} + ( \zeta_{40} - \zeta_{40}^{7} ) q^{78} + ( \zeta_{40} + \zeta_{40}^{3} - \zeta_{40}^{9} + \zeta_{40}^{15} + \zeta_{40}^{17} ) q^{80} + \zeta_{40}^{16} q^{81} + ( \zeta_{40}^{6} + \zeta_{40}^{14} ) q^{82} + ( -\zeta_{40}^{13} + \zeta_{40}^{19} ) q^{83} + ( -\zeta_{40} + \zeta_{40}^{3} + \zeta_{40}^{7} + \zeta_{40}^{17} ) q^{86} + ( 1 - \zeta_{40}^{2} - \zeta_{40}^{6} + \zeta_{40}^{10} + \zeta_{40}^{14} - \zeta_{40}^{16} ) q^{88} + ( -\zeta_{40}^{7} - \zeta_{40}^{17} ) q^{89} + ( \zeta_{40}^{4} + \zeta_{40}^{18} ) q^{90} + ( 1 - \zeta_{40}^{12} ) q^{94} + ( -\zeta_{40}^{3} - \zeta_{40}^{5} + \zeta_{40}^{9} + \zeta_{40}^{11} - \zeta_{40}^{17} - \zeta_{40}^{19} ) q^{96} + ( -\zeta_{40}^{9} + \zeta_{40}^{15} ) q^{98} + ( -\zeta_{40}^{9} + \zeta_{40}^{11} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 4q^{4} + 4q^{9} + O(q^{10}) \) \( 16q + 4q^{4} + 4q^{9} - 4q^{10} - 16q^{16} - 4q^{30} - 4q^{36} - 4q^{39} - 20q^{40} - 16q^{49} + 4q^{55} + 16q^{64} + 8q^{66} + 4q^{75} - 4q^{81} + 20q^{88} + 4q^{90} + 12q^{94} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/975\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(326\) \(352\)
\(\chi(n)\) \(-1\) \(-1\) \(\zeta_{40}^{4}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
194.1
−0.156434 0.987688i
−0.987688 + 0.156434i
0.987688 0.156434i
0.156434 + 0.987688i
0.891007 + 0.453990i
0.453990 0.891007i
−0.453990 + 0.891007i
−0.891007 0.453990i
0.891007 0.453990i
0.453990 + 0.891007i
−0.453990 0.891007i
−0.891007 + 0.453990i
−0.156434 + 0.987688i
−0.987688 0.156434i
0.987688 + 0.156434i
0.156434 0.987688i
−1.69480 + 0.550672i −0.587785 0.809017i 1.76007 1.27877i 0.156434 + 0.987688i 1.44168 + 1.04744i 0 −1.23134 + 1.69480i −0.309017 + 0.951057i −0.809017 1.58779i
194.2 −0.863541 + 0.280582i 0.587785 + 0.809017i −0.142040 + 0.103198i 0.987688 0.156434i −0.734572 0.533698i 0 0.627399 0.863541i −0.309017 + 0.951057i −0.809017 + 0.412215i
194.3 0.863541 0.280582i 0.587785 + 0.809017i −0.142040 + 0.103198i −0.987688 + 0.156434i 0.734572 + 0.533698i 0 −0.627399 + 0.863541i −0.309017 + 0.951057i −0.809017 + 0.412215i
194.4 1.69480 0.550672i −0.587785 0.809017i 1.76007 1.27877i −0.156434 0.987688i −1.44168 1.04744i 0 1.23134 1.69480i −0.309017 + 0.951057i −0.809017 1.58779i
389.1 −1.16110 1.59811i −0.951057 0.309017i −0.896802 + 2.76007i −0.891007 0.453990i 0.610425 + 1.87869i 0 3.57349 1.16110i 0.809017 + 0.587785i 0.309017 + 1.95106i
389.2 −0.183900 0.253116i 0.951057 + 0.309017i 0.278768 0.857960i −0.453990 + 0.891007i −0.0966818 0.297556i 0 −0.565985 + 0.183900i 0.809017 + 0.587785i 0.309017 0.0489435i
389.3 0.183900 + 0.253116i 0.951057 + 0.309017i 0.278768 0.857960i 0.453990 0.891007i 0.0966818 + 0.297556i 0 0.565985 0.183900i 0.809017 + 0.587785i 0.309017 0.0489435i
389.4 1.16110 + 1.59811i −0.951057 0.309017i −0.896802 + 2.76007i 0.891007 + 0.453990i −0.610425 1.87869i 0 −3.57349 + 1.16110i 0.809017 + 0.587785i 0.309017 + 1.95106i
584.1 −1.16110 + 1.59811i −0.951057 + 0.309017i −0.896802 2.76007i −0.891007 + 0.453990i 0.610425 1.87869i 0 3.57349 + 1.16110i 0.809017 0.587785i 0.309017 1.95106i
584.2 −0.183900 + 0.253116i 0.951057 0.309017i 0.278768 + 0.857960i −0.453990 0.891007i −0.0966818 + 0.297556i 0 −0.565985 0.183900i 0.809017 0.587785i 0.309017 + 0.0489435i
584.3 0.183900 0.253116i 0.951057 0.309017i 0.278768 + 0.857960i 0.453990 + 0.891007i 0.0966818 0.297556i 0 0.565985 + 0.183900i 0.809017 0.587785i 0.309017 + 0.0489435i
584.4 1.16110 1.59811i −0.951057 + 0.309017i −0.896802 2.76007i 0.891007 0.453990i −0.610425 + 1.87869i 0 −3.57349 1.16110i 0.809017 0.587785i 0.309017 1.95106i
779.1 −1.69480 0.550672i −0.587785 + 0.809017i 1.76007 + 1.27877i 0.156434 0.987688i 1.44168 1.04744i 0 −1.23134 1.69480i −0.309017 0.951057i −0.809017 + 1.58779i
779.2 −0.863541 0.280582i 0.587785 0.809017i −0.142040 0.103198i 0.987688 + 0.156434i −0.734572 + 0.533698i 0 0.627399 + 0.863541i −0.309017 0.951057i −0.809017 0.412215i
779.3 0.863541 + 0.280582i 0.587785 0.809017i −0.142040 0.103198i −0.987688 0.156434i 0.734572 0.533698i 0 −0.627399 0.863541i −0.309017 0.951057i −0.809017 0.412215i
779.4 1.69480 + 0.550672i −0.587785 + 0.809017i 1.76007 + 1.27877i −0.156434 + 0.987688i −1.44168 + 1.04744i 0 1.23134 + 1.69480i −0.309017 0.951057i −0.809017 + 1.58779i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 779.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
39.d odd 2 1 CM by \(\Q(\sqrt{-39}) \)
3.b odd 2 1 inner
13.b even 2 1 inner
25.e even 10 1 inner
75.h odd 10 1 inner
325.p even 10 1 inner
975.bi odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 975.1.bi.a 16
3.b odd 2 1 inner 975.1.bi.a 16
13.b even 2 1 inner 975.1.bi.a 16
25.e even 10 1 inner 975.1.bi.a 16
39.d odd 2 1 CM 975.1.bi.a 16
75.h odd 10 1 inner 975.1.bi.a 16
325.p even 10 1 inner 975.1.bi.a 16
975.bi odd 10 1 inner 975.1.bi.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
975.1.bi.a 16 1.a even 1 1 trivial
975.1.bi.a 16 3.b odd 2 1 inner
975.1.bi.a 16 13.b even 2 1 inner
975.1.bi.a 16 25.e even 10 1 inner
975.1.bi.a 16 39.d odd 2 1 CM
975.1.bi.a 16 75.h odd 10 1 inner
975.1.bi.a 16 325.p even 10 1 inner
975.1.bi.a 16 975.bi odd 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(975, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 4 T^{2} + 92 T^{4} - 228 T^{6} + 230 T^{8} - 72 T^{10} + 17 T^{12} - 4 T^{14} + T^{16} \)
$3$ \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
$5$ \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
$7$ \( T^{16} \)
$11$ \( 1 + 16 T^{2} + 97 T^{4} - 32 T^{6} + 150 T^{8} + 32 T^{10} + 12 T^{12} + 4 T^{14} + T^{16} \)
$13$ \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
$17$ \( T^{16} \)
$19$ \( T^{16} \)
$23$ \( T^{16} \)
$29$ \( T^{16} \)
$31$ \( T^{16} \)
$37$ \( T^{16} \)
$41$ \( 1 - 4 T^{2} + 92 T^{4} + 228 T^{6} + 230 T^{8} + 72 T^{10} + 17 T^{12} + 4 T^{14} + T^{16} \)
$43$ \( ( 5 + 5 T^{2} + T^{4} )^{4} \)
$47$ \( 1 - 16 T^{2} + 97 T^{4} + 32 T^{6} + 150 T^{8} - 32 T^{10} + 12 T^{12} - 4 T^{14} + T^{16} \)
$53$ \( T^{16} \)
$59$ \( 1 + 16 T^{2} + 97 T^{4} - 32 T^{6} + 150 T^{8} + 32 T^{10} + 12 T^{12} + 4 T^{14} + T^{16} \)
$61$ \( ( 25 + 25 T^{2} + 10 T^{4} + T^{8} )^{2} \)
$67$ \( T^{16} \)
$71$ \( 1 - 4 T^{2} + 92 T^{4} + 228 T^{6} + 230 T^{8} + 72 T^{10} + 17 T^{12} + 4 T^{14} + T^{16} \)
$73$ \( T^{16} \)
$79$ \( T^{16} \)
$83$ \( 1 - 16 T^{2} + 97 T^{4} + 32 T^{6} + 150 T^{8} - 32 T^{10} + 12 T^{12} - 4 T^{14} + T^{16} \)
$89$ \( ( 16 + 8 T^{2} + 4 T^{4} + 2 T^{6} + T^{8} )^{2} \)
$97$ \( T^{16} \)
show more
show less